Method computes rank of complex Hadamard error correction codes
The rank of $\mathbb{Z}_{2^s}$-linear Hadamard codes
Information Theory
Summary
Hadamard codes help detect and fix errors in digital data. The authors study a special kind called \(\mathbb{Z}_{2^s}\)-linear Hadamard codes, which were well understood only for small cases. They developed a method to calculate the rank, a key property, for all sizes, including very large ones. This rank follows a neat formula with consistent coefficients independent of code length. Their work provides exact formulas for larger code classes beyond previously known limits.
What this means in practice
- •For communications engineers: Design error correction schemes with predictable properties for data transmission using \(\mathbb{Z}_{2^s}\)-linear Hadamard codes of any length.
- •For storage system developers: Utilize explicit rank formulas to optimize code construction for reliable data storage systems requiring robust error correction.
Authors
Xuan Wang, Minjia Shi
Abstract
The canonical $\mathbb{Z}_{2^s}$-linear Hadamard codes are the Gray images of the $\mathbb{Z}_{2^s}$-additive codes. Their rank, the dimension of the binary linear span, was known in closed form only for $s\le 3$, and for $s \ge 4$ only through computer computations for small lengths. We give a method that computes the rank for every $s$. We show that the rank is a binomial polynomial in the type parameters, with coefficients $ρ_s(\boldsymbol{k})$ independent of the length. We compute all the coefficients for $s\le 5$, which yields explicit rank formulas over $\mathbb{Z}_4$ and $\mathbb{Z}_8$ (recovering the known ones), and over $\mathbb{Z}_{16}$ and $\mathbb{Z}_{32}$.